A Heston Model with a Stochastic Long-Run Variance Level
Summary
The document proposes extending the Heston stochastic volatility model by making its long-run variance level itself a mean-reverting stochastic process. The spot variance mean-reverts toward that changing level, while the asset price follows a diffusion with jumps under a risk-neutral specification. The author leaves open how the long-run variance process might relate to other model parameters and asks whether prior literature has studied this structure.
The motivation is option valuation: the author is attempting a Heston- or Bates-style characteristic-function derivation for European options. Introducing the additional state variable appears to produce a more complicated Riccati differential equation, and the author is unsure whether a closed-form solution exists. The document presents a model specification and an unresolved derivation challenge, not a completed pricing method, proof of tractability, empirical assessment, or evidence that the extension improves fit. It is useful as a description of a possible volatility-model extension and its mathematical difficulty.
Key ideas
- The proposed extension makes Heston’s long-run variance level a stochastic, mean-reverting process.
- Spot variance mean-reverts toward the time-varying long-run level.
- The specification also allows jumps in the underlying asset price under the risk-neutral measure.
- Adding the extra variance state complicates the characteristic-function equations used for option pricing.
- The document leaves tractability and empirical usefulness unresolved.
Tags
Full text
# Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?
# Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?
I'm working on my dissertation in Financial Economics, focusing on the topic of Stochastic Volatility Jump Diffusion models; and I'm playing around with some ideas for model extensions. In particular, I am quite interested in the idea of an extension in which the Heston long-run mean instantaneous variance parameter is also stochastic and mean-reverting; i.e. with a (risk-neutral) SDE system of: $$dS_{t}=(r_{t}-q_{t}-\lambda \bar{k})S_{t}dt+\sqrt{\nu_{t}}S_{t}dW_{t}^{S}+kS_{t}dN_{t}$$ $$d\nu_{t}=\kappa_{\nu}(\theta_{t}-\nu_{t})dt+\sigma_{\nu}\sqrt{\nu_{t}}dW_{t}^{\nu}$$ $$d\theta_{t}=\kappa_{\theta}(\theta_{\infty}-\theta_{t})dt+\sigma_{\theta}\sqrt{\theta_{t}}dW_{t}^{\theta}$$ $$d\left<W^{S},W^{\nu}\right>=\rho dt$$ $$N_{t}\sim Pois(\lambda t)$$ $$k=\exp(J^{S})-1$$ $$\bar{k}=\mathbb{E}^{Q}\left[k\right],$$
such that $\theta_{\infty}$, $\kappa_{\theta}$, $\sigma_{\theta}$ are the analogous paramaters of the long-run mean instantaneous variance DE to those of the DE for the spot instantaneous variance $\nu_{t}$. I leave open the questions of the jump size distribution (from which the general solution approach is independent assuming independent jumps) as to whether or not the long-run mean instantaneous variance is correlated with any of the other parameters in the model. (I see no reason why it should have to be.)
My question to you guys: is anyone aware of any literature that deals with such a model specification, either with or without jumps included? I ask because I have tried deriving the European Option price Heston/Bates style that would require a $\theta_{t}$ coefficient in the Characteristic Function, and I end up with a rather nasty Riccati DE for that coefficient that is stretching me beyond the limit of my talent I fear. (Or which may simply have no closed-form solution.)
Alternatively, if any of you are feeling lucky today and would like to help a simple mind like my mine try to derive a solution, that would also be greatly appreciated; let me know, I'll type out the DE for you to take a stab at.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.